This report contains three simulation studies for the Clusterwise Independent Vector Analysis (C-IVA) procedure

simulation 1

The first simulation is contains randomly generated data that is based on a CIVA model. This is a basic ‘concept of proof’ simulation. No special correlation structure between S across scanning sessions is imposed.

\[ X_i^{(D)} = P_{ir} S^{(D)}A_i^T\] Note that D represents a scanning session, other terms are similar to CICA. Note that S values are generated from a multivariate Laplace distribution.

The design is as follow:

fixed:

Size of the data are therefore: 2000 (V) X 50 (Ti) X 3 (D) X 40 subjects (R=2) or 80 subjects (R=4)

20 repetitions were used.

Note that the Ai matrices are non-square, which is not something that you encounter in the IVA literature. This was solved by computing:

\[ A_i^{(D)} = X_i^{(D)}S^{(D)T}(S^{(D)}S^{(D)T})^{\dagger} \]

Where \(\dagger\) is the Moore-Penrose pseudo inverse, this is a similar step that is done in (C-)ICA.

results:

## Summary() of Adjusted Rand Index of clustering:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##       1       1       1       1       1       1
## Summary() of percentage of random starts that yielded the lowest loss function value
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.7333  0.9000  0.9667  0.9488  1.0000  1.0000
## Summary() of S Tucker congruence values:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.6954  0.9903  0.9975  0.9706  0.9984  0.9997
## Summary() of A Tucker congruence values:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.6694  0.9971  0.9984  0.9728  0.9992  0.9999

interactions:

Congruence of S

##    Err Q R Nr Stuck.mean
## 1  0.1 2 2 20  0.9978669
## 2  0.1 2 4 20  0.9991927
## 3  0.1 4 2 20  0.9697199
## 4  0.1 4 4 20  0.9833681
## 5  0.1 6 2 20  0.9105273
## 6  0.1 6 4 20  0.9484611
## 7  0.3 2 2 20  0.9990016
## 8  0.3 2 4 20  0.9976728
## 9  0.3 4 2 20  0.9890821
## 10 0.3 4 4 20  0.9886818
## 11 0.3 6 2 20  0.9257090
## 12 0.3 6 4 20  0.9412026
## 13 0.6 2 2 20  0.9962632
## 14 0.6 2 4 20  0.9978663
## 15 0.6 4 2 20  0.9909513
## 16 0.6 4 4 20  0.9912035
## 17 0.6 6 2 20  0.8998694
## 18 0.6 6 4 20  0.9438293

Interactions S (first panel = 10% noise, middle = 30% noise, third panel 60% noise). X axis represents components, split by clusters (colours)

I guess CIVA robust against noise (as was the case for CICA), difficulties with underlying model complexity (Q) but unexpected results or R effect (more clusters seems to improve results).

Congruence of A

##    R Q Nr Err Atuck.mean
## 1  2 2 20 0.1  0.9992391
## 2  2 2 20 0.3  0.9994163
## 3  2 2 20 0.6  0.9960557
## 4  2 4 20 0.1  0.9864100
## 5  2 4 20 0.3  0.9806526
## 6  2 4 20 0.6  0.9878275
## 7  2 6 20 0.1  0.8914611
## 8  2 6 20 0.3  0.9400483
## 9  2 6 20 0.6  0.8793592
## 10 4 2 20 0.1  0.9995025
## 11 4 2 20 0.3  0.9985120
## 12 4 2 20 0.6  0.9987863
## 13 4 4 20 0.1  0.9922047
## 14 4 4 20 0.3  0.9985119
## 15 4 4 20 0.6  0.9889148
## 16 4 6 20 0.1  0.9690951
## 17 4 6 20 0.3  0.9307943
## 18 4 6 20 0.6  0.9738912

Interactions A (first panel = 10% noise, middle = 30% noise, third panel 60% noise). X axis represents components, split by clusters (colours)